72 



THE ORBIT OF URANUS. 



The derivations with respect to y and the node have been omitted because they 

 are quite insensible. The terms of ^R depending on these derivatives are given 

 by equation (31). In the case of Uranus disturbed by Saturn the largest values 

 of the coefficients 



ii/tcotiy; Ihtanly,^ 



6h^ 



da 



are only about .05, while the largest coefficients in hli\ 5//, and hy are less than 10". 

 Hence the largest terms in (31) will be of the order of magnitude 0".5 multiplied 

 by the mass of Saturn, and may therefore be omitted entirely. Omitting them 



the values of hR, h 





OR 



dv 



dR 



dp 



and 



dR 



dv 



d^ 



d\- 



6 —z — become 

 <9p 



dR 



^'^+4?-^^'+i7^p 



dR 



dp 



^V 



. ,rT-R d'-R . ,d'R. 



dvdv 



dvdp 



dvdp dVdp dp'' ' 



dvdp' 

 d-R 



dpdp 



/¥ 



All the separate factors from which the second members of these equations 

 are formed have already been given. Forming tlieir products in the way described 

 in Chapter II, we have the result given in the following tables. 



The expressions for ^R are arranged so that the value of D'tSR can be obtained 

 from them by direct differentiation. This is done by distinguishing the time 

 introduced into R by the co-ordinates of Uranus from that introduced by the 

 co-ordinates of Saturn. 



